The dead yaw opens into a wedge

A yaw that is singular everywhere found the defect these essays have been following: a paired Gough platform held level is singular at every position in its workspace at one yaw, because at that orientation its six leg moments are equal and the leg screws become dependent. The dead yaw is a design choice found that rotating the platform’s pairs by $\rho$ puts that yaw at $90° - \rho$. Tilted, near the dead yaw tilted the platform at the dead yaw and found that the everywhere-singular orientation becomes an ordinary singular surface through the workspace, reachable at every tilt tried.

That last essay looked along one line in orientation space — the dead yaw, tilted — and ended by asking about the rest of it. The dead yaw is one point in a three-dimensional space of orientations. Around it, which orientations leave a singularity somewhere in the workspace? If they form a surface, a task can be told to avoid a yaw; if they fill a region, it has to be told to avoid a neighbourhood, and the size of that neighbourhood becomes a specification.

They fill a region, and its shape is simple enough to state as a rule.

An instrument that does not stall #

The earlier essay measured closeness to a singularity with the smallest singular value of the six leg screws, and found out the hard way that a search downhill on it can stall: a descent from one start halted short of nought and suggested a threshold that did not exist, and forty-eight starts were needed to find the surface.

The question here can be put so that no search is needed at all. The determinant of the six leg screws is nought exactly at a singularity and has a sign everywhere else, and the sign says which side of the singular set a pose is on. A workspace contains a singular surface, at a given orientation, exactly when the determinant takes both signs over it. And for one position, the determinant as a function of yaw near the dead one crosses nought once: at the yaw where that position is singular. That root can be found by bisection, to as many figures as the arithmetic allows, with nothing to stall. Each evaluation is cheap, because a Gough platform’s inverse problem is six square roots: given a pose, every leg’s line is known outright, and the six screws and their determinant follow with no iteration anywhere.

So the set of orientations whose workspace contains a singularity is, at each tilt, the range of those roots over the workspace. The workspace is the earlier essays’ box: horizontal offsets within ±0.8 of home and heights of 1.8, 2.4 and 3.2, sampled at 243 positions.

A root of a determinant could in principle be a sign change through something other than a singularity. It is not here. At every root checked, the smallest singular value is at the arithmetic’s floor, below $10^{-7}$, and a hundredth of a degree of yaw either side it is above $10^{-6}$. The two instruments agree on where the singularities are; the determinant simply finds them without searching.

Level, a single yaw #

The first question is what the set looks like on the level slice, where the earlier essays did all their work.

Level, it is the dead yaw and nothing else. A quarter of a degree below it the determinant is negative at all 243 positions, and a quarter of a degree above it is positive at all 243. The platform is singular everywhere at the dead yaw and nowhere within a quarter of a degree of it, and at a tenth of a degree the same holds. On the level slice the bad set really is a point on the yaw axis, which is why the earlier essays could describe it as one orientation — the spatial counterpart of the planar platform in two orientations no position can rescue, singular at an orientation and not at a place.

Tilt the platform one degree and that stops. A fifth of a degree either side of the dead yaw the determinant already takes both signs over the workspace — seven positions positive and 236 negative on one side, 240 and three on the other. The band of yaws that leave a singular surface in the workspace has opened, and it keeps opening as the tilt grows.

Each position is singular at its own yaw #

What opens the band is visible in the first figure. Level, every position is singular at the same yaw. Tilted, each position is singular at its own, and the pattern across the workspace is a plane.

At half a degree of tilt about $x$, the singular yaw times the position’s height fits

$$\delta\, z \approx -0.155\,x - 0.157\,y - 0.003$$

over all 243 positions, with the worst miss 0.7% of the spread. Without the height — fitting the yaw itself to a plane — the worst miss is 23%, so the height is part of the law and not a correction to it. The three heights fall on one line. Tilt about $y$ instead and the plane turns: the coefficients become $+0.155$ and $-0.162$, and in general the plane’s slope turns with the tilt axis, degree for degree, as the first figure’s slider shows.

In words: a tilt moves each position’s singular yaw by an amount proportional to the tilt and to the position’s horizontal offset in one direction, divided by its height. The direction is set by the tilt axis. Near home the offset is nought, and the constant term is small because home’s singular yaw does not move to first order at all. It moves as the square of the tilt — by 0.0012°, 0.0049° and 0.0193° at half a degree, one and two — and that is exactly the quadratic rise the earlier essay measured there. The earlier measurement was right about home and home is the one place where the answer is second order.

That also explains the earlier essay’s derivation, which borrowed its reasoning from why the platform stays flat and in which the symmetry of the paired arrangement protected the dead yaw from a tilt to first order. It does, at home, where the arrangement is symmetric about the tilt. A position off to one side is not on the symmetry plane, the protection is lost there, and the singular yaw moves at first order. A symmetry that protects a point protects only the point.

The band opens linearly #

The range of singular yaws over the workspace is then the range of a plane over a box, and a plane’s range grows in proportion to its slope — which is to say, linearly with the tilt.

It does. At one degree of tilt the band’s half-width is 0.268° of yaw for tilt about $x$, 0.217° about the diagonal and 0.279° about $y$. Halving the tilt halves the band to within 2% on all three axes and at two rotations of the pairs; a quadratic law, the one home obeys, would quarter it and misses by more than a fifth.

The axis matters a little and for a reason the plane law gives directly. The band’s half-width is the plane’s largest value over the workspace, which falls at a corner of the box at the lowest height, and whether the plane’s slope points along a diagonal of the box or along an edge decides how far out the largest offset is. With the slope along a diagonal the reach is $0.8\sqrt 2$ and the half-width $0.31 \times 1.13 / 1.8 = 0.28°$ per degree; along an edge it is 0.8 and the prediction falls to 0.20°, against a measured 0.217°.

Past about four degrees the two edges of the band part company. At eight degrees about $x$ the band runs from −1.98° to +1.19°, and tilted eight degrees the other way it runs from −5.12° to +3.50°. The linear law is a small-tilt law, and by eight degrees the terms it leaves out are as large as the terms it keeps. Four degrees of tilt and a degree of yaw is roughly where it stops being a good description.

A wedge standing on a point #

Putting the pieces together gives the shape of the set the earlier essay asked about.

Cut the three-dimensional space of orientations through the dead yaw by the plane of yaw and tilt about one axis. Level, the set is a single point. Away from level its two edges open linearly, one on each side. So the set is a wedge standing on its point — a region of orientation space whose thickness in yaw grows in proportion to the tilt — and the same is true in every direction of tilt, with a slope that varies by about a quarter between directions. It is not a surface: any tilt at all, however small, makes a whole interval of yaws bad.

That answers the first of the two questions the earlier essay left, and the answer is useful. “Avoid the dead yaw” cannot be written as a condition on yaw alone, because at a tilt the bad yaws are an interval. It can be written as a condition on orientation: keep the yaw’s distance from the dead yaw more than about 0.28 times the tilt, for this workspace. And because the law is a plane over the workspace with the height underneath, the constant is a property of the machine and its task together — horizontal reach over the lowest working height — rather than of the machine alone. A task that works higher or closer to home needs a narrower margin in yaw, in proportion.

The margin belongs to the task #

The plane law makes a prediction about the workspace that can be tested directly: the band’s width is the plane’s range over the positions the task uses, so shrinking the task should shrink the band in proportion.

It does, on both counts. At one degree of tilt about $x$, the full workspace’s band is 0.535° wide. A task confined to half the horizontal reach — offsets within ±0.4 rather than ±0.8, at the same three heights — has a band 0.268° wide, 0.50 of the full one. A task that works only at the top height, 3.2, has a band of 0.308°, 0.58 of the full one against the 0.56 that the ratio of the lowest heights, 1.8 to 3.2, predicts. Neither comparison uses the plane: both are fresh bisections at every position of the smaller workspace.

So the forbidden neighbourhood of the dead yaw is not a property of the machine. The dead yaw is — it is fixed by the legs’ attachment points and by $ho$ — but how far from it a tilted task must stay is fixed by where the task puts the platform. A task at three degrees of tilt over the full box must keep its yaw out of an interval from −0.79° to +0.67° about the dead yaw; the same task over half the reach, out of an interval half as wide. The map of where it fails showed that a workspace is not a shape anybody chooses, but the part of it a task uses is, and here that choice buys yaw.

The rotation chooses where, not how wide #

The earlier essay’s second question was whether the shape of the bad set depends on $\rho$, the rotation between the pairs that chooses where the dead yaw sits. If it did, choosing $\rho$ would become a trade between putting the dead yaw far from a task’s working orientations and keeping the neighbourhood around it small.

It barely does. Across rotations of 0° to 120° and the three tilt axes, the band at one degree of tilt is between 0.434° and 0.592° wide. The dead yaw moves by 120° over that range of rotations, exactly as $90° - \rho$ says, and the wedge travels with it at nearly the same size. So the choice of $\rho$ remains the single-quantity choice it was: put the dead yaw where the task is not. The width of the margin the task then needs is set by the workspace and the tilt, not by the rotation.

What a specification can now say #

The three earlier essays told a designer where the dead yaw is and that tilting does not remove it. This one adds the rule that turns those into something a controller can check before a pose is commanded:

• Level, the forbidden set is one yaw, $90° - \rho$, and a hair either side of it is safe everywhere.
• Tilted by $t$, the forbidden set is the yaws within about $0.28\,t$ of that one — for this workspace — and the margin scales with the workspace’s horizontal reach over its lowest height.
• Within the band, whether a particular position is singular is decided by the plane: the position is singular at the yaw its offset and height say, and the controller can compute that yaw rather than search for it.

That last point is the practical gain from the plane law. An ordinary Gough platform’s direct singularities — including those where legs meet — have to be mapped by sampling; near the dead yaw the map is two coefficients. Tilted, near the dead yaw needed a many-start descent to confirm that a singular surface was present; the plane says where it is, to first order, from two coefficients.

What the wedge does not cover #

One workspace box. The band’s width is the plane’s range over this box, and the plane itself is a statement about positions within it. Positions far outside it — far from home or near the base — were not checked, and the plane law is a first-order law in the offset as well as in the tilt.

Calibration would not find it. A platform that measures itself runs its identification where the machine is well held, and the wedge is by construction where it is not; the plane’s coefficients would have to come from the nominal geometry.

One platform, with its pairs at one spread. The paired arrangement here has base pairs 25° apart and platform pairs 40° apart. The earlier essays found the dead yaw’s position independent of the spreads; whether the plane’s slope is has not been measured.

The coefficient is measured, not derived. The slope of 0.31° of yaw per degree of tilt per unit of offset over height is read from the fit. The derivation in the earlier essay explains why home escapes the first-order term; a derivation of the coefficient elsewhere, from the leg moments, would make the rule exact to first order rather than fitted.

Still open: a path that has to cross #

A task whose yaw has to pass from one side of the dead yaw to the other has to cross the wedge. Level, it crosses at one instant, and at that instant the platform is singular everywhere in its workspace. Tilted, it spends a whole interval of yaw inside the wedge, and throughout that interval a singular surface sweeps across the workspace, passing through each position at that position’s own yaw.

The distinct argument would be which crossing is less exposed. A level crossing is singular at every position for an instant; a tilted one is singular at each position for an instant too — the moment the sweeping surface passes it — but at different instants for different positions. For a platform that must hold one position while it turns, the second might be no better and no worse; for one moving along a path while it turns, the plane law says the surface and the path can be arranged to miss each other, so that the path crosses the dead yaw with the surface always somewhere else. The measurement would be the smallest singular value along combined paths of position and orientation that cross the dead yaw at several tilts, compared with the same paths crossing level, to find whether a tilted crossing can be made singularity-free when a level one cannot.